How to display a WordPress Custom Field only on a specific day of the week?

You are not checking the current date correctly. You need to use current_time() funciton to get current datetime according to WordPress configuration. Then, using PHP date() you can check what day of week current date is:

// Get current date time of WordPress in Unix timestamp format
$timestamp = current_time( 'timestamp' );

// Get day of the week of current date
// See http://php.net/manual/es/function.date.php
$dw = date( "w", $timestamp);

// Check day of the week: 0 for Sunday to 6 for Saturday
if( $dw == 0 ) {

    // Display content if day of the week is Sunday

}